eyes not moving 100% with head / eyes lagging behind
I took a look at your file. Rigging eyes is complicated. I think the problem had to do with either:

The eye bones not sitting directly on the origin of the eyeballs... 

OR

..that the head bone (spine.004) Y axis didn't sit on the same location as the origin of the eye balls but I'm not sure which or if any of those were what were causing the eyes to yaw off target. It always gets really complicated when you're working with many different bone rotations and origin points etc.

OR

That the eyes were a single object and needed separating. As the front of the head moved from side to side, the eyes only moved in accordance with the rotation of the head bone tail and not at the same height and angle as the eye sockets.

Now, I think it works perfectly but you let me know if it doesn't work.

If you want to follow a good tutorial on it, I use a similar method to this guy. This video is great :)
https://www.youtube.com/watch?v=-9GDj8_bx6M

Here you can download your edited rig:
https://www.dropbox.com/s/5xbykg8brg5msn2/human_male_EDIT.blend?dl=0

Let me know if you can download that file and I'll upload it to another sharing website.

Hope that helps :)